# 基于模糊PID的路径跟踪控制系统Path Tracking Control System Based on Fuzzy PID

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In order to improve the performance of the path tracking control system of the automatic naviga-tion transplanter, a control method based on the fuzzy PID algorithm to adjust the steering angle of the front wheel of the transplanter is proposed. Firstly, the kinematic model of the controlled object is established, and the fuzzy PID control algorithm is designed according to the kinematic model. Secondly, Matlab is used to simulate the path tracking control system. The results show that the performance of the fuzzy PID control method is better than the traditional PID control method, which can effectively reduce the overshoot of the system and the time to reach the steady state. Finally, the dynamic test is carried out to verify that under the traditional PID control method, the maximum path tracking error is 3.8 and 6.5 cm, and the average path tracking error is 2.47 and 3.67 cm when the car chassis is running at the speed of 0.35 and 0.85 m/s; under the fuzzy PID control method, the maximum path tracking error is 2.1 and 4.8cm, and the average path tracking error is 1.57 and 2.7 cm when the car chassis is running at the speed of 0.35 and 0.85m/s. The experiment shows that the performance of fuzzy PID control is better than that of traditional PID algorithm, and it is more suitable for path tracking control of transplanter.

1. 引言

2. 控制系统总体结构

Figure 1. Path tracking control principle

3. 运动学模型

Figure 2. Kinematic model of transplanter

$\left\{\begin{array}{l}{x}^{\prime }\left(t\right)=v\mathrm{sin}\theta \\ {y}^{\prime }\left(t\right)=v\mathrm{cos}\theta \\ {\theta }^{\prime }\left(t\right)=\frac{v\mathrm{tan}\delta }{L}\end{array}$ (1)

v：插秧机速度；

$\theta$ ：航向角，取逆时针方向为正；

$\delta$ ：前轮转角，取逆时针方向为正。

4. 模糊PID控制系统

4.1. PID控制

PID控制采用比例、积分和微分的方式对系统进行控制。PID控制分为模拟PID控制和数字PID控制两种方式，本研究用离散化的方法对输入信号进行处理，采用数字PID控制方式来实现对系统的控制 [14]。传统PID控制表示方法：

$u\left(t\right)={K}_{p}\left[e\left(t\right)+\frac{1}{{T}_{i}}{\int }_{0}^{t}e\left(t\right)\text{d}t+{T}_{d}\frac{\text{d}e\left(t\right)}{\text{d}t}\right]$ (2)

${K}_{p}$ ：比例系数；

$e\left(t\right)$ ：输入信号；

${T}_{i}$ ：积分时间；

${T}_{d}$ ：微分时间。

$\Delta u\left(k\right)={k}_{p}\left(e\left(k\right)-e\left(k-1\right)\right)+{k}_{i}e\left(k\right)+{k}_{d}\left(e\left(k\right)-2e\left(k-1\right)+e\left(k-2\right)\right)$ (3)

$e\left(k\right)$ ：输入偏差；

$e\left(k-1\right)$ ：上一次输入偏差；

$e\left(k-2\right)$ ：上两次输入偏差。

4.2. PID控制的参数整定

Table 1. Adjustment rules of critical scale method parameters

4.3. 模糊PID控制设计

${K}_{p}={K}_{p1}+\Delta {K}_{p}$ (4)

${K}_{i}={K}_{i1}+\Delta {K}_{i}$ (5)

${K}_{d}={K}_{d1}+\Delta {K}_{d}$ (6)

Figure 3. Structure of fuzzy PID system

1) 输入、输出变量的模糊化

a) 横向偏差e

b) 偏差变化率ec

c) 比例系数∆Kp

d) 积分系数∆Ki

e) 微分系数∆Kd

2) 定义输入和输出隶属度函数

Figure 4. Membership function of input variables e and ec

Figure 5. Membership function of ∆Kp, ∆Ki, ∆Kd output variables

3) 建立模糊控制规则

a) 横向偏差e和偏差变化率ec的正负表示方向，当横向偏差e的绝对值较大时，系统应快速减小横向偏差e，为提高系统的反应速率，应设置较大的Kp值；提高系统的反应速率会产生微分溢出，为防止微分溢出，应设置较小的Kd值；同时为防止出现积分饱和现象，应设置Ki的值为0。

b) 当横向偏差e的绝对值较小时，为保证系统稳定性良好，Kp值和Ki值应取较大的值；同时为增强路径跟踪控制系统的抗干扰能力，应该选取适当的Kd值，Kd值的具体调整原则：当偏差变化率ec的绝对值较小时，Kd值应取较大值，当偏差变化率ec的绝对值较大时，Kd值应取较小值。

Table 2. Kp fuzzy control rules

Table 3. Ki fuzzy control rules

Table 4. Kd fuzzy control rules

Figure 6. ∆Kp output surface

Figure 7. ∆Ki output surface

Figure 8. ∆Kd output surface

4) 模糊推理和反模糊化

Table 5. Fuzzy logic reasoning parameter setting

$u=\frac{\int u\ast {\mu }_{N}\left(u\right)\text{d}u}{\int {\mu }_{N}\left(U\right)\text{d}u}$ (7)

$u=\frac{\sum {u}_{i}\ast {\mu }_{N}\left({u}_{i}\right)}{\sum {\mu }_{N}\left({u}_{i}\right)}$ (8)

5. 系统仿真与试验验证

5.1. 仿真分析

Figure 9. Simulation response curve of control system

5.2. 动态试验验证

Table 6. Main parameters of trolley chassis

Figure 10. Error statistics under PID control method

Figure 11. Error statistics under Fuzzy PID control method

6. 结论

1) 提出一种基于模糊PID控制算法的自动导航插秧机路径跟踪控制方法。该模糊PID控制相比于传统PID控制算法可减小调整超调量，减少响应时间，可提高系统动态性能。

2) 通过试验验证表明，在模糊PID控制方法下，当小车底盘以0.35和0.85 m/s的速度行驶时，最大路径跟踪误差为2.1和4.8 cm，平均路径跟踪误差为1.57和2.7 cm。验证了模糊PID控制方法性能更优，且对低速和高速都具有适应性。

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